here its Lyapunov characteristic exponents are \(\lambda_{1,2}=-\rho \pm i \omega\ ,\) \(\rho>0\ ,\) \(\omega \neq 0\ ,\) and \(\lambda_3=\gamma>0\ ;\) smooth functions \( F_i\ ,\) along with their first
partials, vanish at the origin. The stable manifold \(W^s\) of the saddle-focus \(O\) is 2D then, whereas the unstable one \(W^u\) is 1D. The manifold \(W^u\) is the union of \(O\) and two separatrices that tend to \(O\) as \(t\to-\infty\ .\) A homoclinic loop \(\Gamma\) of the saddle-focus is a trajectory bi-asymptotic to \(O\) as \(t \to \pm \infty\ ,\) see Figure 1. In other words, \(\Gamma \in W^s \cap W^u\ .\)

Saddle value

Next we introduce the saddle value or quantity \(\sigma=-\rho+\gamma\) and the saddle index \(\nu=\rho/\gamma\ .\) Depending on the sign of \(\sigma\ ,\) or whether \(\nu\) is less or greater than \(1\ ,\) the dynamics of (1) near the saddle-focus is simple when \(\sigma<0\ ,\) or complex if \(\sigma>0\ .\)

The analysis of a homoclinic bifurcation is reduced to that of a Poincarè mapping defined by
the trajectories nearby the homoclinic loop \(\Gamma\ .\) For sake of simplicity we assume that the above system (1) is linear near the origin.

Global Mapping

First, in a small neighborhood of the saddle-focus the cross-section \(\Pi_1\) is introduced as a transverse to the stable manifold \(W^s_{loc}\) breaking \(\Pi_1\) into the top \(\Pi^+_1\) and bottom \(\Pi^-_1\) components. The homoclinic loop hits \(\Pi_1\) at some point \(M^* \in \Gamma \cap \Pi_1\) on \(W^s\ .\)

Next we need to determine the image of \(\Pi_1\) on the second cross-section \(\Pi_2\) transverse to \(W^u_{loc}\ ;\) it will have the shape of a spiral with countably many revolutions accumulating to \(W^u_{loc}\ .\) Thus, the local map \(T_{0}\) from \(\Pi_1 \to \Pi_2\) is defined.
Its properties are solely determined by the characteristic exponents of the saddle-focus.

We can now evaluate the flight time \(\tau = -\frac{1}{\gamma} \ln \frac{z_0}{d}\) of the trajectory connecting the cross-sections. Clearly, that this time increases logarithmically fast the closer the initial point is to the stable manifold \(W^s_{loc}\ .\) Substituting \(\tau\) into the first equation of (2) gives
the local map \(T_0: \Pi_1^+ \mapsto \Pi_2\) along the trajectories passing by near \(O\ :\)
\[\tag{3}
\begin{array}{rcl}
x_1 &=& x_0 \left(\frac{z_0}{d}\right)^\nu
\cos \frac{\omega}{\gamma} \ln \frac{d}{z_0},\\ \\
y_1 &=& x_0 \left(\frac{z_0}{d}\right)^\nu
\sin \frac{\omega}{\gamma} \ln \frac{d}{z_0}.
\end{array}
\]

Observe that the map \(T_0\) is defined only for \(z_0>0\ ,\) because the forward trajectory of an initial point on \(\Pi_1^{-}\) below the stable manifold \(W^s_{loc}\) leaves the saddle-focus and never comes back, unless there is a re-injection, for example, due to the occurrence of a simultaneousness or symmetric
second homoclinic loop, like in the case illustrated in Figure 7.

We can see from (3) that the image \(T_0 \Pi_0^+\) on \(\Pi_1\) looks like a "snake"
which spirals onto the point \(M^-=\Gamma\cap \Pi_1\ ,\) Fig. (Figure 2).

The image of \(\Pi_0^+\) under \(T_1\circ T_0\) preserves the spiraling shape too. It intersects \(W^s_{\rm loc}\) infinitely many times around \(M^+\ ,\) as shown in Fig. Figure 2.

Let us strip the upper section \(\Pi_1^+\) down into a countable number \(k\) of the segments \(\Sigma_k\ .\) The image \(T\Sigma_k\) of is one half of a single curl of the "snake". It follows from (2) and (5) that the top of the \(k\)-th curl is estimated as \[ z \sim z_{2k}^\nu \sim e^{-2\pi\nu k/\omega}.\]
Thus, when \(\nu>1\ ,\) there is no intersection between \(\Sigma_k\) and
\(T\Sigma_k\) as the image of \(\Sigma_k\) is below its pre-image.
On the contrary, when \(\nu<1\ ,\) the intersection
\(T\Sigma_k\cap \Sigma_k\) is non-empty and consists of two connected
components (Fig. Figure 2). It is geometrically evident that there is a
fixed point of the return mapping \(T\) within each of the components. A contemporary
reader may observe that this leads to a formation of the topological Smale horseshoe [1963], which became a
de facto proof of dynamical chaos nowadays. Note that a fixed point of the
Poincarè first return map (5) corresponds to a periodic orbit of
the system (1).

Figure 2: Poincarè mapping on the cross-section \(\Pi_1\) is a contraction (left) when the saddle value \(\sigma<0\) (\(\nu>1\)), or an expansion (right) if \(\sigma>0\) \((\nu<1)\) where the non-empty intersection \(T\Sigma_k\cap \Sigma_k\) generates the Smale horseshoes.

Shilnikov theorem

Theorem [L. Shilnikov, 1965] If the saddle index \(\nu < 1\ ,\) i.e. the saddle value \(\sigma>0\ ,\) then there are countably many saddle periodic orbits in a neighborhood of the homoclinic loop \(\Gamma\) of the saddle-focus.

The condition \(\nu<1\) also known as the Shilnikov condition is imperative here,
because the structure of the phase space near the homoclinic loop in case \(\nu>1\) is trivial and leads only to the emergence of a single, stable periodic orbit from the homoclinic loop.

The bifurcations in the boundary case \(\nu=1\ ,\) where small perturbations trigger the system between the homoclinic explosion (\(\nu<1\)) and the trivial dynamics (\(\nu>1\)), were first considered in [L.Belyakov, 1973].

We see from it that when \(\nu<1\) there are infinitely many roots accumulating to zero:
\[\tag{8}
z^*_k = C e^{-\pi k \gamma/ \omega}(1 + o(1)) \quad \mbox{as} \quad k \to + \infty,
\]

where \(C = e^{(\theta - \frac{\pi}{2})\gamma/\omega}\ .\)

Some properties of of system with a saddle-focus can be revealed through
examining the following one-dimensional Poincarè mapping
\[\tag{9}
z_{n+1} = \mu+A z_{n}^{\nu} \cos\left(\frac{\omega}{\gamma} \ln\frac{1}{z_n} + \theta\right) + \cdots\,,
\]

where \(\mu\) is the bifurcation parameter controlling the decomposition of the primary
homoclinic loop existing at \(\mu=0\ .\) The graph of this mapping is shown in Fig. Figure 3 in both cases.

Figure 3: 1D Poincarè mapping (9) in cases \(\nu>1\) and \(\nu<1\ .\) It is clearly seen that in the latter case there are countably many fixed points near the primary homoclinic loop \(\Gamma\) at \(\mu=0\ .\) As this parameter is varied, they undergo saddle-node and period doubling bifurcations, while the system generates more complex homoclinic loops.

The systems with homoclinic loops of a saddle-focus form a bifurcation manifold \(B^1\) of codimension one
in the Banach space of dynamical systems with a smooth topology. Hence, small smooth perturbations
of the original vector field break the homoclinic loop down in general. Then, in case \(\nu>1\ ,\) the
transition over \(B^1\) leads to the emergence of a single stable periodic orbit from the homoclinic loop.
This means that the given bifurcation remains (1) in the Morse-Smale class of systems with simple
dynamics.

In contrast, the feature of case \(\nu <1\) is complex dynamics. As Fig. Figure 2 shows that starting
with some \(k>\bar k\ ,\) the image \(T\Sigma_k\) crosses the pre-image \(\Sigma_k\) twice like the Smale horseshoe. Therefore, \(T\Sigma_k \cap \Sigma_k=\Sigma_k^1 \cup \Sigma_k^2\) contains a hyperbolic set \(\Omega_k\)
homeomorphic to a Bernoulli subshift on two symbols. However, the nonwandering set
\(\bigcup_{\bar k=k}^{\infty} \Omega_k\) is comprised not only of the saddle periodic orbits that are entirely in a neighborhood of the homoclinic loop. Some other saddle trajectories exist here as well due to the jumps between distinct strips. For example, for a jump from \(\Sigma_i\) to \(\Sigma_j\) one needs the fulfillment of the inequality \(j < \nu\,i\ .\) This makes the saddle index \(\nu\) be nothing else but a topological invariant, called an \(\omega\)-modulus. Its changes lead to transformations in the structure of the nonwandering set near the primary homoclinic loop. So, alone with [dis]appearances of hyperbolic subsets there occur bifurcations of periodic orbits as well as formations of homoclinic tangencies between the stable and unstable manifolds of the saddle periodic orbits, etc. Thus, the occurrence of a homoclinic loop of a saddle focus with \(\nu<1\) is a basic criteria of complex dynamics in any system.

Topological types of saddle-foci

Speaking of nonlinear dynamics applications, we need to distinguish the following three cases of saddle-foci in
high-dimensional systems by their leading characteristic exponents, Fig. Figure 4:

As one can notice from Fig. Figure 3 that variations of the control parameter \(\mu\) give rise to period-doubling and saddle-node bifurcations of periodic orbits in the vicinity of the primary homoclinic loop of the saddle-focus.
One must then wonder: under what conditions does the system itself and close ones have
no stable periodic orbits near the homoclinic bifurcation? If there is at least one characteristic exponent in the right half-plane, then
the answer is positive by default. Otherwise, in case of where all other characteristic exponents of the saddle are further to left from the imaginary axes, we need to introduce the second saddle value, respectively in all three cases, as follows:

Figure 4: Three saddle-foci are categorized by their leading characteristic exponents. It is supposed that there no other characteristic exponents with positive real parts.

Thus, if \(\sigma_2 >0\)e, the system has no stable orbits near on \(B^1\)
[Ovsyannikov and L. Shilnikov, 1987 and 1992, Glendenning and Sparrow, 1984]. On the contrary,
when \(\sigma_2<0\ ,\) then the systems with stable periodic orbits are dense
everywhere on \(B^1\ .\)

Figure 5: (left) Homoclinic saddle-focus of the third type or a bi-focus. (right) The geometry of the corresponding Poincarè mapping (from L.Shilnikov et at., 1998).

Routes to Shilnikov/spiral chaos

Next let us consider a bifurcation scenario leading to the formation of a spiral attractor.
This scenario is de-facto proven to be very typical for a plethora of models, including the Lorenz-84 model
[A.Shilnikov et al, 1994] that will illustrate it here. Initially, let a system \(X_\mu\) have a stable equilibrium state \(O\) for \(\mu<\mu_1\ .\) Next \(O\) undergoes a supercritical Andronov-Hopf bifurcation at \(\mu=\mu_1\) so that a stable periodic orbit
\(L_\mu\) bifurcates from \(O\ .\) This orbit becomes the boundary of the unstable manifold \(W^u_{loc}\) of the new saddle-focus for \(\mu>\mu_1\) that spirals onto \(L_\mu\ ,\) Fig. Figure 6(A). Suppose that as \(\mu\) increases further, the multipliers of \(L_{\mu}\) cross a unit circle outward. Then the periodic orbit becomes unstable being enveloped by a new born stable 2D torus for \(\mu>\mu_2\ .\) The unstable manifold of the saddle-focus spirals
onto the torus now thereby increasing the size of the a whirlpool (Fig. Figure 6(B)). Next let \(W^u\) touch \(W^s\ :\) this forms the aforementioned homoclinic loop \(\Gamma\) of the saddle-focus (of the
second type (Fig. Figure 6(C))). After that the attracting whirlpool will contain a set of
complex structure - the so-called spiral strange attractor. If the second saddle value \(\sigma_2<0\ ,\) this spiral
attractor is indeed a quasi-chaotic attractor [L. Shilnikov, 1981] as itself or one in a close system may have stable periodic orbits in it. However, if \(\sigma_2>0\) than the behavior of trajectories in the whirlpool is hyperchaotic. The onset of the spiral attractor can be preceded by the breakdown of the 2D torus, or by a period doubling cascade.

Figure 6: Stages of the formation of a spiral attractor in the Lorenz-84 model (from A.Shilnikov et al, 1995).

The wavetrain generated by the spiral attractor has the distinct shape, Fig. (Figure 7).
It has the quiescence periods, when the phase point comes close by the saddle-focus,
followed by the bursts of oscillatory activity. This signature of the spiral attractor allows for a clear identification of the Shilnikov bifurcation not only in numerical but experimental studies as well, including nonlinear laser optics, various electronic circuits, economics, biology, hydrodynamics flows and many other various fields, just Google this bifurcation.

Figure 7: (left) Bursts generated by the spiral attractor in Figure 7. (right) Dependence of the period of the periodic orbit as it becomes the homoclinic loop in the Chua circuit with the cubic nonlinearity (from L.Shilnikov et al, 2001).

Wild spiral attractor

Finally, we describe a construction [L.Shilnikov and D.Turaev, 1998] allowing for
the occurrence of a wild pseudo-hyperbolic strange attractor in the 4D+ phase space. The construction
is centered around the saddle-focus with two symmetric or simultaneous homoclinic loops \(\Gamma_1\) and
\(\Gamma_2\) Figure 8. Let the saddle-focus have the leading characteristic exponents \(\lambda_{1,2}=-\rho \pm i \omega\ ,\)
\(\lambda_{3}= \gamma>0\ ,\) all other ones are on the left from \(\lambda_{1,2}\) in the complex plane. We require that:

(A) both saddle values are positive\[\sigma_1=-\rho+\gamma>0\] and \(\sigma_2=-2\rho+\gamma>0\ ;\)

The essence of the last property is that the Poincarè
mapping shall possess an invariant foliation. Having factorized over the
leaves of the foliation we can create a factor-mapping that expands phase volumes. This means that no stable
orbit of the system itself or of a close one can exist in the domain \(D\ .\) Such an attractor is also wild in sense
of Newhouse [Newhouse, 1974] because the periodic orbits in it have various homoclinic tangencies.
The fractal dimension of this chain-transitive attractor exceeds three.

Figure 8: (left) A system with the absorbing domain \(\Pi_1\cup \Pi_0 \cup \Pi_2\) containing two homoclinic loops of a saddle-focus in \(R^4\) may generate a wild, pseudo-hyperbolic attractor (right).

References

Shilnikov L. P. [1965] A case of the existence of a denumerable set of periodic motions, Sov. Math. Dokl. 6, 163–166.

Shilnikov L. P. [1967] The existence of a denumerable set of periodic motions in four-dimensional space in an extended neighborhood of a saddle-focus, Soviet Math. Dokl.8(1), 54–58.

Shilnikov L. P. [1970] A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type, Math. USSR Sbornik10, 91–102.